Short answers, pulled from the story.
Thomas William Rhys Davids first translated the Pali term sati into English as mindfulness in 1881. This translation appeared on page 107 of his work Buddhist Suttas published by Clarendon Press.
The Sanskrit term smriti literally means that which is remembered. Monier Monier-Williams's dictionary lists eight distinct meanings for this term including remembrance, reminiscence, and memory.
John D. Dunne an associate professor at University of Madison-Wisconsin asserts that translating these terms as mindfulness is confusing. He suggests establishing retention as a preferred alternative translation.
Originally mindfulness provided way to liberation by paying attention to sensory experience preventing arising of disturbing thoughts emotions causing further chain reactions leading to rebirth. Paul Williams refers to Erich Frauwallner stating mindfulness provides constant watching sensory experience preventing arising cravings powering future experience into rebirths.
Rupert Gethin observes that contemporary Vipassana movement interprets Satipatthana Sutta as describing pure form of insight meditation where samatha and jhana are unnecessary. This interpretation contrasts with views from Georges Dreyfus who stresses Buddhist context includes remembering indicating function includes retention of information.